Chapter 9

Trigonometric Equations

Equations in which the trig ratios of unknown angles are involved are called "Trigonometric Equations" such as the ones shown below:

sinx + cosx = 0.75    and    tan2x - 3tanx = 0.

The solution (s) to a trig equation are angles that satisfy the equation.  For example, one of the solutions to the trig equation:  cos2x + sinx = 1 is x = π/6.  To verify, let's substitute π/6 for x in the equation as follows:

cos (π/3) + sin(π/6) = 1    or,

1/2 + 1/2 =1    or,

1 = 1.

To solve the trig equations, we keep simplifying them until they reduce to one of the following forms:

sinx = sinβ,    or    cosx = cosβ,    or,    tanx = tanβ, and we use the method of Chapter 8, to write down the possible solution sets.

Example 1:

 Solve the trig equation:  sinx + tanx cos(2π - x) = 1.

Example 2: Solve the trig equation given below and find all solutions between 0 and 2π.

2sin2x + 3cosx = 3.

We got two possible sets of answers for each cosine equation; however, in set (1):

x = k(2π) ± 0 is just x = 2kπ.

In set (2): we may write it as x = 2kπ ± π/3.

The problem is asking for answers between 0 and 2π.

Setting k = 0, (1) yields: x = 0, and (2) yields: x = +π/3.  Note that x = -π/3 is out of range.

Setting k = 1, (1) yields: x = 2π, and (2) yields: x = 2π-π/3.    x = 2π +π/3 is out of range.

Finally, the collection of the desired answers are: x = 0, π/3, 5π/3, and 2π.

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Example 3: Solve the trig equation given below and find all solutions between 0 and 2π.

tanx + cotx = 2.

For k = 0, the solution is x = π/4.

For k = 1, the solution is x = 5π/4.

For any other k value, the solution will be out of the desired range.

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Exercises: Solve the following trigonometric equations: